- Email: [email protected]
Solitons stabilization in PT symmetric potentials through modulation the shape of imaginary component
Optics & Laser Technology 88 (2017) 104–110
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
crossmark
Solitons stabilization in PT symmetric potentials through modulation the shape of imaginary component ⁎
Chunfu Huang , Jiale Zeng College of Physics and Materials Science, Tianjin Normal University, Tianjin, 300387 China
A R T I C L E I N F O
A BS T RAC T
Keywords: Optical solitons PT symmetry Propagation Stability
Stability and dynamics of PT symmetric fundamental bright solitons supported by localized potentials in a focusing Kerr medium are investigated numerically. How the shape and magnitude of the imaginary component affect soliton stability is addressed when fixed real part of the potentials. The unbroken PT symmetry in linear case and stable region in nonlinear case are discussed. Numerical simulations proved that solitons can propagate stably when the loss or gain distribution become narrower. So we can stabilize the solitons through modulation the shape of imaginary component, and can use the method for solitons formation and control in other nonlinear media with PT potentials.
1. Introduction Stabilization of beams in media is a very important problem in nonlinear optics. Spatial solitons would form when the beams diffraction is balanced by the nonlinearity or lattices potentials. Much investigation show that plenty of optical nonlinearities (cubic, quadratic, photorefractive, nonlocal, etc) and lattices potentials could support spatial solitons [1–3]. Recently, propagation of optical beam in complex nonlinear media featuring the Parity-Time (PT) symmetry has drawn considerable attention [4–41]. The PT symmetry originates from quantum mechanics. It is shown that non-Hermitian Hamiltonians exhibiting PT ^ is symmetry could have entire real spectrum. The parity operator P ^ ^ ^ ^ ^ ^ defined by the relations P →−P , x →−x , where P , x stand for momentum and position operators, respectively. Whereas that of the time ^ by P^ →−P^ , x^ →x^ ,i →−i . One can deduce that, operator Τ 2 ^ ^ ^H ^ =P^ 2 /2 + V * (−x ) = H ^ . Thus a necessary con^T ^ TH=P /2 + VPT * (x ), P PT dition for the Hamiltonians to be PT symmetry is that the potential function VPT (x ) should satisfy the condition VPT * (x ) = VPT * (−x ) [4–6]. So the real part of the PT potential must be a symmetric function of position whereas the imaginary component should be anti-symmetric. PT symmetry may exhibit entirely real spectrum in some parameter regions, which is called the unbroken PT symmetry [7–10]. While in the broken PT symmetry, the spectrum becomes complex and the propagating solitons waves may be destroyed. The connection, bridged by the schrödinger equation, between the quantum mechanics and optical field made it possible to realize such idea in optical structures. Most interesting, the propagation and
⁎
existence of various types of solitons in diverse complex potentials with gain or loss have attracted much attention [11–41]. For example, gap solitons [11–13], vector solitons in parity-time-symmetric lattices [14], gray solitons in parity-time symmetric potentials [15], discrete solitons in self-defocusing systems with PT-symmetric defects [16], lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices [17], PT solitons in integrable nonlocal nonlinear schrödinger equation [18], nonlocal PT gap and defect solitons [19–21], etc. Also several complex potentials have been introduced, for example, periodic potential [22,23], harmonic potential [24,25], Gaussian potential [26], double-delta well and Wadati potentials [27] etc. A recent review about nonlinear waves in PT-symmetric systems is shown in Ref. [28]. It is demonstrated the existence of infinite families of localized solutions with multiple humps and dips between the two potential singularities in double-delta well. The Wadati potentials can support constant density waves. This property has been used to study the modulational instability within the Gross-Pitaevskii equations with complex potentials [29]. In the context of systems with gain and loss, the PT-symmetric Wadati potentials are unique among all PT-symmetric potentials in supporting continuous families of asymmetric solitons [30]. Moreover the stability of PT solitons has been addressed [30–39]. For example, stability analysis for solitons in PT-symmetric optical lattices [31]; All-real spectra in optical systems with arbitrary gainand-loss distributions [32]; Also it is shown that the nonlinearity can soften the PT-symmetry breaking transition in the nonlinearly-coupled dimer [33,34], etc. All these results show soliton formation and stability are very
Corresponding author. E-mail address: [email protected] (C. Huang).
http://dx.doi.org/10.1016/j.optlastec.2016.09.009 Received 26 February 2016; Received in revised form 24 July 2016; Accepted 8 September 2016 0030-3992/ © 2016 Elsevier Ltd. All rights reserved.
Optics & Laser Technology 88 (2017) 104–110
C. Huang, J. Zeng
Fig. 1. (a) Profile of real part of PT potential; (b) profile of imaginary part of PT potential, where V0=1 and W0=1, respectively.
Fig. 2. Soliton profiles for m=1 (red dot), 5 (green dash), 9 (blue solid) at (a) μ=1 and (b) μ=7 when W0=3, respectively. Symmetric curves represent the real part of the profile. Antisymmetric curves represent the imaginary part (imaginary part is multiplied by 10 for μ=7). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. (a) Pictures of broken PT symmetry and unbroken PT symmetry. (b) The dependence of W0 (where soliton looses stability) upon the propagation constant μ for different m. Where m=9,7,5,3,2,1 (from top to bottom), respectively.
all the solitons are actually unstable. In this paper a PT potential with real part V (X ) = V0sech2 (X ) and imaginary part W (X ) = W0 tanh(X )sechm (X ) is considered, where m > 0. In this case the real part is fixed and the imaginary part is varied with m solely. Through linear stability analysis and beam propagation method the stability and dynamics of fundamental bright solitons in such potential are investigated. Moreover the unbroken PT symmetry
sensitive to the PT potential and nonlinearity. In particular the imaginary part play an important role in the formation and propagation of solitons. Quite recently, Jisha et al. discussed the imaginary component effect on the solitons formation and propagation [40,41], in which paper the PT potential with super-gaussian profile are considered, and both real and imaginary part varied simultaneously. Results show that
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Optics & Laser Technology 88 (2017) 104–110
C. Huang, J. Zeng
Fig. 4. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=2 (top row), 7 (bottom row) when W0=3, m=1.
106 Fig. 5. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=2 (top row), 7 (bottom row) when W0=3, m=9.
Optics & Laser Technology 88 (2017) 104–110
C. Huang, J. Zeng
Fig. 6. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=1(top row), 7 (bottom row) when W0=1.25, m=1.
∂ 2ϕ + ϕ 2 ϕ + VPT ϕ = μϕ ∂X 2
in linear case and stable region in nonlinear case are discussed. Results show that, the larger the m, the narrower the loss or gain distribution, and the wider the stable region. So it is easy to stabilize the solitons when increased the parameter m. For a given m, the soliton can be stable/unstable when W0 below/above some critical value of W0c. So the solitons can be stabilized through modulation the shape of imaginary component. All of which can be used for solitons formation and control in other nonlinear media with PT potentials.
In this paper ϕ (X ) is a localized solution with a complex function, and such localized solutions can be obtained with help of many numerical methods. Such as spectral renormalization method [42], modified squared operator method [43], etc. To analyze the linear stability of the localized solutions, we consider a small perturbation to the localized solution of Eq. (1) in the form [4,31,37,40,48]:
2. Theoretical model
ψ (X , Z )
To start, let us assume that the light beams propagate along the Z axis and diffract in the X directions. Propagation of monochromatic light in a (1+1)-dimensional local Kerr nonlinear system involves a PT symmetric complex index distribution obeys a modified nonlinear schrödinger equation [4–6]
i
∂ψ (X , Z ) ∂ 2ψ (X , Z ) + + ψ 2 ψ (X , Z ) + VPT ψ (X , Z ) = 0 ∂Z ∂X 2
=
{ϕ (X ) + ε [ f (X, Z )exp(λZ ) + g* (X, Z )exp(−λ*Z )] }exp(iμZ )
(4)
Substituting the perturbed solution into Eq. (1) and linearizing with respect to ε (ε < < 1), one obtains the following linearized eigenvalue problem:
(1)
L^
where ψ is proportional to the electric field envelope, ψXX term describe the optical diffraction. The PT-symmetric potential VPT = V (X ) + iW (X ). In case of PT-symmetry, the potential satisfies the condition V (X ) = V (−X ) and W (−X ) = −W (−X ). Where the function V (X ) is associated with index guiding, and W (X ) represents the gain(loss) distribution of the potential. Solitons solutions are sought in the following form,
ψ (X , Z ) = ϕ (X )exp(iμZ )
(3)
{} {} f g
=λ
f g
(5)
⎧ ⎫ ⎪ L^1 ϕ2 ⎪ ^ ⎬, L1 = ∂XX + 2 ϕ 2 + VPT − μ. f (X , Z ), g (X , Z ) where L^ = ⎨ ⎪ ^ *⎪ ⎩ − ϕ*2 − L1 ⎭ are the eigenfunctions of the linearized eigenvalue problem. “*” is complex conjugate operator. If real(λ)=0 for every λ (i.e., the system possesses solely image eigenvalues), the solution is linearly stable. If at least one of the eigenvalues is purely real or complex, the soliton is unstable. Here the Fourier collocation method [44] is used to obtain the whole spectrum, and the results are checked numerically by the beam propagation method.
(2)
Where ϕ (X ) is the nonlinear eigenmode. μ being the propagation constant. Substitute Eq. (2) into Eq. (1) we can get the following equation: 107
Optics & Laser Technology 88 (2017) 104–110
C. Huang, J. Zeng
of m. For larger W0, both the real and imaginary parts have a significant change. For larger m, the real part of soliton almost maintain the same shape; whereas the peaks of the imaginary part get narrower and the amplitude decrease a little. According to Fig. 1(b), the gain or loss region is decreased with increase of m, i.e. the larger the m, the narrower the region, so for small μ at larger m, wide solitons strongly overlapping with the gain or loss regions (see Fig. 2(a)). In this potential we considered, when given m, the solitons are not easy to exist for larger W0, which can be explained from Fig. 3(b). It is known PT symmetry may exhibit entirely real spectrum in some parameter regions, which is called the unbroken PT symmetry. While in the broken PT symmetry, the spectrum becomes complex and the solitons beams may be destroyed. Typical pictures of broken and unbroken PT symmetry in linear case are shown in Fig. 3(a). The system have real/complex spectrum when W0 below/above the threshold W0T. The W0T increase with increase of m. At the same time, when considered the nonlinear case, the stability parameter regions may be changed. Typical regions are shown in Fig. 3 (b). It shows the dependence of W0 (where soliton looses stability) on the propagation constant μ for different m. We see that for a given m, the soliton can be stable/unstable when W0 below/above some critical value of W0c. The larger the m, the wider the stable region. When for a lower m, the stable region is narrower. From Fig. 3 it seems that W0c is always less than W0T. In other words, the unbroken PT-symmetry is necessary condition for the stability, but not sufficient. The solitons obtained for larger W0 at low power tend to be strongly unstable, while obtained for lower W0 can be stable. In particular the critical value of W0c is varied with propagation constant μ, i.e., the W0c is increased with increase of μ when m is larger than 3. When m is less than 3, the W0c is increased firstly and then decreased, with increase of
3. Solitons solutions and their stability In the following sections we will present some numerical results to show their properties. Firstly we will consider a PT potential with the real part and imaginary part V (X ) = V0sech2 (X ) W (X ) = W0 tanh(X )sechm (X ), where m > 0. The imaginary part corresponds to the potential discussed in Ref. [45]. When m=1, the PT potential is the complex Scarf II potential, considered in Refs [4,46– 48]; When m=0, the potential under consideration corresponds to the Rose-Morse potential well, considered in Ref [49]. When m is changed, the profile of imaginary component will be modulated. Thus, it is possible to study how the shape of the imaginary component affects solitons evolution and their stability. Typical pictures of real and imaginary part of the PT potential are shown in Fig. 1. Where the amplitude V0=1, W0=1, respectively. From Fig. 1 we see the amplitude and shape of real part remain unchanged, while that of imaginary part varied with m. i.e., the peaks and width of the imaginary part decreased with increase of m. The two peaks getting narrower, and the loss or gain region becomes decreasingly concentrated on the middle of the potential. In another words, the larger the m, the narrower the loss or gain region. Typical profiles of solitons solutions with different m and W0 are depicted in Fig. 2. In accordance with PT symmetry, fundamental solitons have an even real part and an odd imaginary part. For larger μ, the intensity increase and the width decrease. Also we find the imaginary part seems has little effect on the profile of the solitons, which means that the nonlinearity dominant the whole process. But from the following numerical simulations we find the shape of imaginary part play a important role in the solitons stability. For small W0, the amplitude and shape of real part of the solitons almost unchanged, while that of the imaginary part decreased with increase
Fig. 7. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=1 (top row), 7 (bottom row) when W0=1.25, m=2.5.
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μ. For the fixed localized potential V (X ) = sech2 (X ) (V0=1), there exist some critical value of μc, which varied with m and W0. Some numerical results are presented to show their propagation and stability properties. For example, when m=1, the stable region is quite narrower (see Fig. 3(b)), and the propagation and stability of such solitons are shown in Fig. 4, Fig. 6 and top row in Fig. 8, respectively. The left picture is stability spectrum, and the right is the max intensity of solitons propagation with distances. Obviously, the solitons in such cases are unstable. Even no random noises are added at the input, all solitons beams are destroyed after propagation a short distance. Which can be explained from Fig. 3(b) that W0 (see W0=3,1.25,0.6 in Fig. 4, Fig. 6 and Fig. 8, respectively) is larger than W0c. As a result all solitons beams are unstable. Considered a more larger m, i.e. m=9, the stable region is wider (see Fig. 3(b)), both the solitons beams with propagation constants μ=2 and 7 can evolve into stable solitons. Numerical simulations show that when a random noise perturbed with a magnitude of 10% of the solitons at the input, the solitons beams can propagate stably with a long longitudinal distance about 8000, and the intensity of such solitons almost unchanged (see Fig. 5). Which can be explained from Fig. 3(b) also, obviously W0 (see W0=3 in Fig. 5) is less than W0c for all propagation constant μ (see the top line in Fig. 3(b)), as a result the solitons beams are stable. Finally considered a moderate m, i.e. m=2.5 (see Fig. 7 and bottom row in Fig. 8), both the solitons beams perturbed with a random noise can propagate stably with a distance about 8000 too. For the parameters W0=1.25 in Fig. 7 and W0=0.6 in Fig. 8 are less than W0c. But it is unstable when W0=3 and m=2.5, for the parameter W0=3 is greater than W0c. And it can be stable when m=9 and W0=3 which discussed above. From the pictures in Figs. 4–8 we can conclude that, when m is
small, the stable region is quite narrower, so if W0 greater than some critical value of W0c, the eigenvalues of the stability spectrum will be very large, the solution experiences exponential growth due to an instability mode and cannot be stable. As a result the solitons beams propagate with a short distance then be destroyed. When considered a more larger m, the stable region becomes wider, so if W0 below some critical value of W0c, the eigenvalues of the stability spectrum decrease to zeros, which means the soliton solution experiences almost no exponential growth and can evolve into a stable soliton. So it is easy to stabilize the solitons when increased the parameter m for the wider stable regions.
4. Conclusions In conclusions, we have studied the stabilization of PT symmetric solitons in a medium with Kerr nonlinearity through modulation the shape of imaginary component. We considered a PT potential with V (X ) = V0sech2 (X ) and W (X )=W0 tanh(X )sechm (X ), and studied the shape and magnitude of imaginary component affect soliton stability. We discussed the unbroken PT symmetry in linear case and stable region in nonlinear case. Results show that the unbroken PT-symmetry is necessary condition for the stability, but not sufficient, and results also show that the larger the m, the narrower the loss or gain distribution, and the wider the stable region. So it is easy to stabilize the solitons when increased the parameter m. For a given m, the soliton can be stable/unstable when W0 below/above some critical value of W0c. As a result we can stabilize the solitons through modulation the shape of imaginary component, and can use the method for solitons formation and control in other nonlinear media with PT symmetric potentials.
Fig. 8. Stability spectrum (left column) and solitons propagation (right column) for different m=1 (top row), 2.5 (bottom row) when W0=0.6, μ=7.
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Acknowledgments This research was supported by the Foundation for Young Teachers in Tianjin Normal University under Grant no. ZX110QN014. Also supported by the Innovative training program for college students in Tianjin Normal University under Grant no. 201631. References [1] Y.S. Kivshar, G.P. Agrawal, Optical Solitons, Academic, San Diego, 2003. [2] F. Lederer, G.I. Stegemen, D.N. Christodoulides, G. Assanto, M. Segev, Y. Silberberg, Phys. Rep. 463 (2008) 1–126. [3] Y.V. Kartashov, B.A. Malomed, L. Torner, Rev. Mod. Phys. 83 (1) (2011) 247–305. [4] Z.H. Musslimani, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Phys. Rev. Lett. 100 (2008) 030402. [5] K.G. Makris, R. El-Ganainy, D.N. Christodoulides, Z.H. Musslimani, Phys. Rev. Lett. 100 (2008) 103904. [6] Z.H. Musslimani, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, J. Phys. A: Math. Theor. 41 (2008) 244019. [7] C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243. [8] C.M. Bender, Rep. Prog. Phys. 70 (2007) 947. [9] G. Levai, M. Znojil, J. Phys. A 33 (2000) 7165. [10] Z. Ahmed, Phys. Lett. A282 (2001) 343. [11] X. Zhu, H. Wang, L. Zheng, Opt. Lett. 36 (14) (2011) 2680–2682. [12] L. Dong, L. Gu, D. Guo, Phys. Rev. A 91 (2015) 053827. [13] L. Ge, M. Shen, C. Ma, Opt. Exp. 22 (24) (2014) 29435–29444. [14] Y.V. Kartashov, Opt. Lett. 38 (14) (2013) 2600–2603. [15] H. Li, Z. Shi, X. Jiang, Opt. Lett. 36 (16) (2011) 3290–3292. [16] Z. Chen, J. Huang, J. Chai, X. Zhang, Y. Li, B.A. Malomed, Phys. Rev. A 91 (2015) 053821. [17] Y. He, X. Zhu, M. Dumitru, Phys. Rev. A. 85 (1) (2012) 013831. [18] (a) M.J. Ablowitz, Z.H. Musslimani, Phys. Rev. Lett. 110 (2013) 064105; (b) M.J. Ablowitz, Z.H. Musslimani, Nonlinearity 29 (2016) 915–946. [19] S. Hu, X. Ma, D. Lu, Phys. Rev. A 85 (4) (2012) 043826. [20] H. Wang, W. He, S. Shi, Phys. Scri. 89 (2) (2014) 025502. [21] C.P. Jisha, A. Alessandro, V.A. Brazhnyi, Phys. Rev. A. 89 (1) (2014) 013812.
110
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
crossmark
Solitons stabilization in PT symmetric potentials through modulation the shape of imaginary component ⁎
Chunfu Huang , Jiale Zeng College of Physics and Materials Science, Tianjin Normal University, Tianjin, 300387 China
A R T I C L E I N F O
A BS T RAC T
Keywords: Optical solitons PT symmetry Propagation Stability
Stability and dynamics of PT symmetric fundamental bright solitons supported by localized potentials in a focusing Kerr medium are investigated numerically. How the shape and magnitude of the imaginary component affect soliton stability is addressed when fixed real part of the potentials. The unbroken PT symmetry in linear case and stable region in nonlinear case are discussed. Numerical simulations proved that solitons can propagate stably when the loss or gain distribution become narrower. So we can stabilize the solitons through modulation the shape of imaginary component, and can use the method for solitons formation and control in other nonlinear media with PT potentials.
1. Introduction Stabilization of beams in media is a very important problem in nonlinear optics. Spatial solitons would form when the beams diffraction is balanced by the nonlinearity or lattices potentials. Much investigation show that plenty of optical nonlinearities (cubic, quadratic, photorefractive, nonlocal, etc) and lattices potentials could support spatial solitons [1–3]. Recently, propagation of optical beam in complex nonlinear media featuring the Parity-Time (PT) symmetry has drawn considerable attention [4–41]. The PT symmetry originates from quantum mechanics. It is shown that non-Hermitian Hamiltonians exhibiting PT ^ is symmetry could have entire real spectrum. The parity operator P ^ ^ ^ ^ ^ ^ defined by the relations P →−P , x →−x , where P , x stand for momentum and position operators, respectively. Whereas that of the time ^ by P^ →−P^ , x^ →x^ ,i →−i . One can deduce that, operator Τ 2 ^ ^ ^H ^ =P^ 2 /2 + V * (−x ) = H ^ . Thus a necessary con^T ^ TH=P /2 + VPT * (x ), P PT dition for the Hamiltonians to be PT symmetry is that the potential function VPT (x ) should satisfy the condition VPT * (x ) = VPT * (−x ) [4–6]. So the real part of the PT potential must be a symmetric function of position whereas the imaginary component should be anti-symmetric. PT symmetry may exhibit entirely real spectrum in some parameter regions, which is called the unbroken PT symmetry [7–10]. While in the broken PT symmetry, the spectrum becomes complex and the propagating solitons waves may be destroyed. The connection, bridged by the schrödinger equation, between the quantum mechanics and optical field made it possible to realize such idea in optical structures. Most interesting, the propagation and
⁎
existence of various types of solitons in diverse complex potentials with gain or loss have attracted much attention [11–41]. For example, gap solitons [11–13], vector solitons in parity-time-symmetric lattices [14], gray solitons in parity-time symmetric potentials [15], discrete solitons in self-defocusing systems with PT-symmetric defects [16], lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices [17], PT solitons in integrable nonlocal nonlinear schrödinger equation [18], nonlocal PT gap and defect solitons [19–21], etc. Also several complex potentials have been introduced, for example, periodic potential [22,23], harmonic potential [24,25], Gaussian potential [26], double-delta well and Wadati potentials [27] etc. A recent review about nonlinear waves in PT-symmetric systems is shown in Ref. [28]. It is demonstrated the existence of infinite families of localized solutions with multiple humps and dips between the two potential singularities in double-delta well. The Wadati potentials can support constant density waves. This property has been used to study the modulational instability within the Gross-Pitaevskii equations with complex potentials [29]. In the context of systems with gain and loss, the PT-symmetric Wadati potentials are unique among all PT-symmetric potentials in supporting continuous families of asymmetric solitons [30]. Moreover the stability of PT solitons has been addressed [30–39]. For example, stability analysis for solitons in PT-symmetric optical lattices [31]; All-real spectra in optical systems with arbitrary gainand-loss distributions [32]; Also it is shown that the nonlinearity can soften the PT-symmetry breaking transition in the nonlinearly-coupled dimer [33,34], etc. All these results show soliton formation and stability are very
Corresponding author. E-mail address: [email protected] (C. Huang).
http://dx.doi.org/10.1016/j.optlastec.2016.09.009 Received 26 February 2016; Received in revised form 24 July 2016; Accepted 8 September 2016 0030-3992/ © 2016 Elsevier Ltd. All rights reserved.
Optics & Laser Technology 88 (2017) 104–110
C. Huang, J. Zeng
Fig. 1. (a) Profile of real part of PT potential; (b) profile of imaginary part of PT potential, where V0=1 and W0=1, respectively.
Fig. 2. Soliton profiles for m=1 (red dot), 5 (green dash), 9 (blue solid) at (a) μ=1 and (b) μ=7 when W0=3, respectively. Symmetric curves represent the real part of the profile. Antisymmetric curves represent the imaginary part (imaginary part is multiplied by 10 for μ=7). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. (a) Pictures of broken PT symmetry and unbroken PT symmetry. (b) The dependence of W0 (where soliton looses stability) upon the propagation constant μ for different m. Where m=9,7,5,3,2,1 (from top to bottom), respectively.
all the solitons are actually unstable. In this paper a PT potential with real part V (X ) = V0sech2 (X ) and imaginary part W (X ) = W0 tanh(X )sechm (X ) is considered, where m > 0. In this case the real part is fixed and the imaginary part is varied with m solely. Through linear stability analysis and beam propagation method the stability and dynamics of fundamental bright solitons in such potential are investigated. Moreover the unbroken PT symmetry
sensitive to the PT potential and nonlinearity. In particular the imaginary part play an important role in the formation and propagation of solitons. Quite recently, Jisha et al. discussed the imaginary component effect on the solitons formation and propagation [40,41], in which paper the PT potential with super-gaussian profile are considered, and both real and imaginary part varied simultaneously. Results show that
105
Optics & Laser Technology 88 (2017) 104–110
C. Huang, J. Zeng
Fig. 4. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=2 (top row), 7 (bottom row) when W0=3, m=1.
106 Fig. 5. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=2 (top row), 7 (bottom row) when W0=3, m=9.
Optics & Laser Technology 88 (2017) 104–110
C. Huang, J. Zeng
Fig. 6. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=1(top row), 7 (bottom row) when W0=1.25, m=1.
∂ 2ϕ + ϕ 2 ϕ + VPT ϕ = μϕ ∂X 2
in linear case and stable region in nonlinear case are discussed. Results show that, the larger the m, the narrower the loss or gain distribution, and the wider the stable region. So it is easy to stabilize the solitons when increased the parameter m. For a given m, the soliton can be stable/unstable when W0 below/above some critical value of W0c. So the solitons can be stabilized through modulation the shape of imaginary component. All of which can be used for solitons formation and control in other nonlinear media with PT potentials.
In this paper ϕ (X ) is a localized solution with a complex function, and such localized solutions can be obtained with help of many numerical methods. Such as spectral renormalization method [42], modified squared operator method [43], etc. To analyze the linear stability of the localized solutions, we consider a small perturbation to the localized solution of Eq. (1) in the form [4,31,37,40,48]:
2. Theoretical model
ψ (X , Z )
To start, let us assume that the light beams propagate along the Z axis and diffract in the X directions. Propagation of monochromatic light in a (1+1)-dimensional local Kerr nonlinear system involves a PT symmetric complex index distribution obeys a modified nonlinear schrödinger equation [4–6]
i
∂ψ (X , Z ) ∂ 2ψ (X , Z ) + + ψ 2 ψ (X , Z ) + VPT ψ (X , Z ) = 0 ∂Z ∂X 2
=
{ϕ (X ) + ε [ f (X, Z )exp(λZ ) + g* (X, Z )exp(−λ*Z )] }exp(iμZ )
(4)
Substituting the perturbed solution into Eq. (1) and linearizing with respect to ε (ε < < 1), one obtains the following linearized eigenvalue problem:
(1)
L^
where ψ is proportional to the electric field envelope, ψXX term describe the optical diffraction. The PT-symmetric potential VPT = V (X ) + iW (X ). In case of PT-symmetry, the potential satisfies the condition V (X ) = V (−X ) and W (−X ) = −W (−X ). Where the function V (X ) is associated with index guiding, and W (X ) represents the gain(loss) distribution of the potential. Solitons solutions are sought in the following form,
ψ (X , Z ) = ϕ (X )exp(iμZ )
(3)
{} {} f g
=λ
f g
(5)
⎧ ⎫ ⎪ L^1 ϕ2 ⎪ ^ ⎬, L1 = ∂XX + 2 ϕ 2 + VPT − μ. f (X , Z ), g (X , Z ) where L^ = ⎨ ⎪ ^ *⎪ ⎩ − ϕ*2 − L1 ⎭ are the eigenfunctions of the linearized eigenvalue problem. “*” is complex conjugate operator. If real(λ)=0 for every λ (i.e., the system possesses solely image eigenvalues), the solution is linearly stable. If at least one of the eigenvalues is purely real or complex, the soliton is unstable. Here the Fourier collocation method [44] is used to obtain the whole spectrum, and the results are checked numerically by the beam propagation method.
(2)
Where ϕ (X ) is the nonlinear eigenmode. μ being the propagation constant. Substitute Eq. (2) into Eq. (1) we can get the following equation: 107
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of m. For larger W0, both the real and imaginary parts have a significant change. For larger m, the real part of soliton almost maintain the same shape; whereas the peaks of the imaginary part get narrower and the amplitude decrease a little. According to Fig. 1(b), the gain or loss region is decreased with increase of m, i.e. the larger the m, the narrower the region, so for small μ at larger m, wide solitons strongly overlapping with the gain or loss regions (see Fig. 2(a)). In this potential we considered, when given m, the solitons are not easy to exist for larger W0, which can be explained from Fig. 3(b). It is known PT symmetry may exhibit entirely real spectrum in some parameter regions, which is called the unbroken PT symmetry. While in the broken PT symmetry, the spectrum becomes complex and the solitons beams may be destroyed. Typical pictures of broken and unbroken PT symmetry in linear case are shown in Fig. 3(a). The system have real/complex spectrum when W0 below/above the threshold W0T. The W0T increase with increase of m. At the same time, when considered the nonlinear case, the stability parameter regions may be changed. Typical regions are shown in Fig. 3 (b). It shows the dependence of W0 (where soliton looses stability) on the propagation constant μ for different m. We see that for a given m, the soliton can be stable/unstable when W0 below/above some critical value of W0c. The larger the m, the wider the stable region. When for a lower m, the stable region is narrower. From Fig. 3 it seems that W0c is always less than W0T. In other words, the unbroken PT-symmetry is necessary condition for the stability, but not sufficient. The solitons obtained for larger W0 at low power tend to be strongly unstable, while obtained for lower W0 can be stable. In particular the critical value of W0c is varied with propagation constant μ, i.e., the W0c is increased with increase of μ when m is larger than 3. When m is less than 3, the W0c is increased firstly and then decreased, with increase of
3. Solitons solutions and their stability In the following sections we will present some numerical results to show their properties. Firstly we will consider a PT potential with the real part and imaginary part V (X ) = V0sech2 (X ) W (X ) = W0 tanh(X )sechm (X ), where m > 0. The imaginary part corresponds to the potential discussed in Ref. [45]. When m=1, the PT potential is the complex Scarf II potential, considered in Refs [4,46– 48]; When m=0, the potential under consideration corresponds to the Rose-Morse potential well, considered in Ref [49]. When m is changed, the profile of imaginary component will be modulated. Thus, it is possible to study how the shape of the imaginary component affects solitons evolution and their stability. Typical pictures of real and imaginary part of the PT potential are shown in Fig. 1. Where the amplitude V0=1, W0=1, respectively. From Fig. 1 we see the amplitude and shape of real part remain unchanged, while that of imaginary part varied with m. i.e., the peaks and width of the imaginary part decreased with increase of m. The two peaks getting narrower, and the loss or gain region becomes decreasingly concentrated on the middle of the potential. In another words, the larger the m, the narrower the loss or gain region. Typical profiles of solitons solutions with different m and W0 are depicted in Fig. 2. In accordance with PT symmetry, fundamental solitons have an even real part and an odd imaginary part. For larger μ, the intensity increase and the width decrease. Also we find the imaginary part seems has little effect on the profile of the solitons, which means that the nonlinearity dominant the whole process. But from the following numerical simulations we find the shape of imaginary part play a important role in the solitons stability. For small W0, the amplitude and shape of real part of the solitons almost unchanged, while that of the imaginary part decreased with increase
Fig. 7. Stability spectrum (left column) and solitons propagation (right column) for different propagation constants μ=1 (top row), 7 (bottom row) when W0=1.25, m=2.5.
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μ. For the fixed localized potential V (X ) = sech2 (X ) (V0=1), there exist some critical value of μc, which varied with m and W0. Some numerical results are presented to show their propagation and stability properties. For example, when m=1, the stable region is quite narrower (see Fig. 3(b)), and the propagation and stability of such solitons are shown in Fig. 4, Fig. 6 and top row in Fig. 8, respectively. The left picture is stability spectrum, and the right is the max intensity of solitons propagation with distances. Obviously, the solitons in such cases are unstable. Even no random noises are added at the input, all solitons beams are destroyed after propagation a short distance. Which can be explained from Fig. 3(b) that W0 (see W0=3,1.25,0.6 in Fig. 4, Fig. 6 and Fig. 8, respectively) is larger than W0c. As a result all solitons beams are unstable. Considered a more larger m, i.e. m=9, the stable region is wider (see Fig. 3(b)), both the solitons beams with propagation constants μ=2 and 7 can evolve into stable solitons. Numerical simulations show that when a random noise perturbed with a magnitude of 10% of the solitons at the input, the solitons beams can propagate stably with a long longitudinal distance about 8000, and the intensity of such solitons almost unchanged (see Fig. 5). Which can be explained from Fig. 3(b) also, obviously W0 (see W0=3 in Fig. 5) is less than W0c for all propagation constant μ (see the top line in Fig. 3(b)), as a result the solitons beams are stable. Finally considered a moderate m, i.e. m=2.5 (see Fig. 7 and bottom row in Fig. 8), both the solitons beams perturbed with a random noise can propagate stably with a distance about 8000 too. For the parameters W0=1.25 in Fig. 7 and W0=0.6 in Fig. 8 are less than W0c. But it is unstable when W0=3 and m=2.5, for the parameter W0=3 is greater than W0c. And it can be stable when m=9 and W0=3 which discussed above. From the pictures in Figs. 4–8 we can conclude that, when m is
small, the stable region is quite narrower, so if W0 greater than some critical value of W0c, the eigenvalues of the stability spectrum will be very large, the solution experiences exponential growth due to an instability mode and cannot be stable. As a result the solitons beams propagate with a short distance then be destroyed. When considered a more larger m, the stable region becomes wider, so if W0 below some critical value of W0c, the eigenvalues of the stability spectrum decrease to zeros, which means the soliton solution experiences almost no exponential growth and can evolve into a stable soliton. So it is easy to stabilize the solitons when increased the parameter m for the wider stable regions.
4. Conclusions In conclusions, we have studied the stabilization of PT symmetric solitons in a medium with Kerr nonlinearity through modulation the shape of imaginary component. We considered a PT potential with V (X ) = V0sech2 (X ) and W (X )=W0 tanh(X )sechm (X ), and studied the shape and magnitude of imaginary component affect soliton stability. We discussed the unbroken PT symmetry in linear case and stable region in nonlinear case. Results show that the unbroken PT-symmetry is necessary condition for the stability, but not sufficient, and results also show that the larger the m, the narrower the loss or gain distribution, and the wider the stable region. So it is easy to stabilize the solitons when increased the parameter m. For a given m, the soliton can be stable/unstable when W0 below/above some critical value of W0c. As a result we can stabilize the solitons through modulation the shape of imaginary component, and can use the method for solitons formation and control in other nonlinear media with PT symmetric potentials.
Fig. 8. Stability spectrum (left column) and solitons propagation (right column) for different m=1 (top row), 2.5 (bottom row) when W0=0.6, μ=7.
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